Adaptive Control of Uncertain Hamiltonian Multi-Input Multi-Output Systems : With Application to Spacecraft Control

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1 28 Americn Control Conference Wetin Settle Hotel, Settle, Whington, USA June 11-13, 28 ThB16.3 Adptive Control of Uncertin Hmiltonin Multi-Input Multi-Output Sytem : With Appliction to Spcecrft Control Hyungjoo Yoon Brij N. Agrwl Abtrct A novel dptive control lw for nonliner Hmiltonin Multi-Input Multi-Output (MIMO) ytem with uncertin prmeter in the ctutor modeling well the inerti /or the Corioli centrifugl term i developed. The phyicl propertie of the Hmiltonin ytem re effectively ued in the control deign the tbility nlyi. The number of the prmeter etimte i ignificntly lowered compred to the conventionl dptive control method. A mooth projection lgorithm i pplied to keep the prmeter etimte inide ingulrity-free region. The developed control cheme i pplied for ttitude control of pcecrft with both the inerti the ctutor uncertintie. I. INTRODUCTION We conider multi-input multi-output (MIMO) nonliner Hmiltonin ytem repreented by the econd-order differentil eqution H(q) q+c(q, q) q+g(q) = F F = D(q)u (1) (1b) q R n i the generlized coordinte vector, H R n n i the (ymmetric poitive definite) inerti mtrix, C q i nonliner vector of Corioli centripetl force, g R n i the grvity vector. F R n i the generlized force i generted by control input vector u R m the ctutor mtrix D R n m. For full trcking control, it i generlly required tht n m D h full row rnk. Expreing dynmic of ytem in the form of Eq. (1), rther thn the tte-pce form, h everl dvntge. Eqution (1) cn be eily derived by pplying Lgrnge eqution, it form i o generl tht it cn repreent vriou kind of dynmic ytem, uch multilink robot mnipultor [1], [2] pcecrft [3] [5], etc. In ddition, there i phyicl property tht the mtrix Ḣ 2C i kewymmetric. Thi property i extremely ueful in deigning dvnced control cheme. Suppoe now tht the ytem mtrice hve uncertintie in their prmeter cn be expreed H(q,Θ ) = H n (q)+h (q,θ ) C(q, q,θ ) = C n (q, q)+c (q, q,θ ) g(q,θ ) = g n (q)+g (q,θ ) D(q,Θ ) = D n (q)+d (q,θ ) (2) (2b) (2c) (2d) the mtrice with upercript of n re with their known nominl vlue, Θ R p i vector of unknown H. Yoon B.N. Agrwl re with the Dept. of Mechnicl Atronuticl Engineering, Nvl Potgrdute School, Monterey, CA 93943, USA. Emil : (See grwl@np.edu bounded contnt uncertintie in the ctutor mtrix D, Θ R q i in the other ytem mtrice/vector, H,C, g. We lo ume tht, by proper definition of the unknown prmeter Θ Θ, the uncertin mtrice H (q,θ ),C (q, q,θ ),g (q,θ ) D (q,θ ) re linerly dependent on Θ, Θ, repectively. Adptive control for pecil ce the ctutor modeling doe not hve uncertintie, tht i D =, h been intenively tudied in the literture (ee for intnce Ref. [1]). However, dptive control for more generl ce with D doe not eem to hve received much ttention in the literture, even though thi uncertinty my reult in ignificnt degenertion of controller performnce. Ge [6] h derived n dptive control lw for multilink mnipultor ytem with uncertintie in the control input term, but the uncertinty mut be in the input cling, thu the uncertinty mtrix mut be digonl when repreented in multiplictive form. (Or it cn be id tht D = D n i digonl mtrix.) Chng [7] h provided n dptive, robut trcking control lgorithm for nonliner MIMO ytem which i bed on the mooth projection lgorithm, which h lo been ued in [8] [9] for dptive control of SISO ytem. More recently, one of the uthor [1], [11] h lo provided n dptive control cheme bed on the mooth projection lgorithm which i pplied to pcecrft ttitude trcking with uncertin milignment/inerti of the ctutor flywheel. However, thee previou reult [7], [1], [11] re bed on purely mthemticl pproche do not exploit the ueful phyicl propertie of the Hmiltonin ytem. More ignificntly, they conidered MIMO ytem repreented by the differentil eqution without ny term in front of the highet derivtive of the tte vrible vector, like the tte-pce form. Therefore, in their method, Eq. (1) would need to be converted q = H 1 C q H 1 g+h 1 Du, (3) in which the uncertin mtrice re multiplied to ech other. Therefore, in order to deign dptive lw bed on the liner dependency of uncertintie, their method need over-prmeteriztion, which men they need to etimte the combintion of the element in Θ Θ, thu the number of prmeter etimte would ignificntly incree. In the preent pper, n dptive control lgorithm for the generl ce the uncertin mtrice in Eq. (2) re ll nonzero i developed uing the mooth projection lgorithm, which keep the prmeter etimte inide properly defined ingulr-free convex et. The propoed dptive lw exploit the phyicl propertie of the Hmiltonion ytem /8/$ AACC. 2969

2 h more compct form with mller number of prmeter etimte. The propoed method i then pplied for pcecrft ttitude control problem with inerti/ctutor uncertintie. Finlly, numericl exmple with the pcecrft re provided to vlidte the propoed lw. II. ADAPTIVE CONTROL LAW The firt prt of the derivtion of the dptive control lw follow the trd deign procedure for Hmiltonin ytem in [1], [4]. Let u ume the deired trjectory q d (t), q d (t) q d (t) to be bounded. The trcking error vector i defined q q q d meure of trcking the reference velocity q r re defined q+λ q = q q r, (4) q r q d Λ q, (5) the mtrix Λ i umed to be Hurwitz. Let ˆΘ be the prmeter etimte vector let Θ ˆΘ Θ be prmeter etimte error vector, when i or. A uggeted by Slotine et. l. [1] from phyicl inight tht q T H q i the ytem kinetic energy, the following Lypunov function cidte i defined V(t) = 1 [ T H+ Θ T Γ 1 Θ + Θ T Γ 1 ] Θ (6) 2 Γ,Γ re poitive definite weighting mtrice. Differentiting V(t) with repect to time uing the kew-ymmetry property of the mtrix Ḣ 2C to replce the term 1 2 T Ḣ with T C, one cn hve the following expreion: V = T (Du H q r C q r g)+ Θ T Γ 1 Θ + Θ T Γ 1 Θ. (7) When the ctutor mtrix D i exctly known (i.e., D = ) h full row rnk (i.e., rnk(d) = n), one cn eily deign n dptive control lw uing the method propoed in the previou work [1]. However, ince D i umed to contin unknown prmeter well H,C g, novel control lw i propoed follow: ( (D n + ˆD )u = (H n + Ĥ ) q r +(C n +Ĉ ) q r ) (8) +(g n + ĝ ) K d the mtrice/vector with ht ymbol re contructed uing the prmeter etimte ˆΘ ˆΘ inted of the (unknown) ctul prmeter. K d i gin mtrix which i poitive definite. When the mtrix D n + ˆD i umed to hve full row rnk, the (weighted) minimum norm olution i given by u = (D n + ˆD ) ( (H n + Ĥ ) q r +(C n +Ĉ ) q r ) (9) +(g n + ĝ ) K d ( ) denote the (weighted) peudo-invere of mtrix [5], [11]. The control lw (9) led to V [{ } = T (D n + ˆD ) D ) u (H n + H ) q r (C n +C ) q r ] (g n + g ) + Θ T Γ 1 Θ + Θ T Γ 1 Θ [ = T (H n + Ĥ ) q r +(C n +Ĉ ) q r +(g n + ĝ ) K d ] D u (H n + H ) q r (C n +C ) q r (g n + g ) + Θ T Γ 1 Θ + Θ T Γ 1 Θ [ ] = T H q r + C q r + g K d D u + Θ T Γ 1 Θ + Θ T Γ 1 Θ = T K d + T ( H q r + C q r + g ) T D u + Θ T Γ 1 Θ + Θ T Γ 1 Θ (1) Since the uncertinty mtrice/vector re umed to depend linerly on Θ, one cn define known regreor mtrice (in fct, row vector) Y = Y (q, q, q r, q r,) R 1 q Y = Y (q, q, q r, q r,,u) R 1 p uch tht T (H q r +C q r + g ) = Y Θ (11) T D u = Y Θ (12) Notice tht the definition of the regreor mtrix Y in Eq. (11) i lightly different from tht in the previou work by Slotine [1], [3]. By including in the regreor definition, the ize of the regreor Y become mller thn tht of Slotine work the regreor h ize of n q. The time derivtive of V then become V = T K d +Y Θ +Y Θ + Θ T Γ 1 ˆΘ + Θ T Γ 1 ˆΘ (13) tking the dpttion lw of the prmeter etimte to be ˆΘ = Γ Y T (14) ˆΘ = Γ Y T (15) then yield V(t) = T K d. (16) Uing trd rgument in [1], [11] which ue Brblt lemm, one cn eily how tht V thu t. Thi lo implie the trcking error q well. A. Smooth Projection Algorithm We previouly ued n umption tht the mtrix D n + ˆD h full row rnk in deriving the dptive control lw, Eq.(9),(14) (15). In generl, for the full trcking control, the nominl mtrix D n in generl h full row rnk. However, drift of the prmeter etimte ˆΘ, governed by n updte lw (15), cn reult in D n + ˆD loing rnk. We will refer to thi itution ingulrity of the teering lw due to the dpttion. Thi ingulrity hinder the ue of the derived control lw, o they need to be modified. 297

3 If the nominl mtrix D n h full row rnk, the true vlue of the prmeter uncertinty Θ i bounded by ufficiently mll number, the prmeter etimte ˆΘ i lo kept mll, then the mtrix D n + ˆD will lo hve full row rnk. To thi end, we define the following two convex et, Ω Θ {Θ R p Θ 2 < β } (17) ˆΩ Θ { ˆΘ R p ˆΘ 2 < β + δ} (18) β > δ > re known contnt. Notice tht Ω Θ ˆΩ Θ. We mke the following three umption. Aumption 1. The nominl vlue D n h full row rnk of n. Aumption 2. The ctul vlue Θ belong to the et Ω Θ. Aumption 3. If ˆΘ ˆΩ Θ, then D n + ˆD i noningulr. Thee umption llow u to modify the dpttion lw (15) by uing the mooth projection lgorithm follow 1 : ˆΘ = Proj( ˆΘ,Φ ) (19) Φ Γ Y T (2) Φ, if (i) ˆΘ 2 < β,or (ii) ˆΘ 2 β Φ T ˆΘ, Proj( ˆΘ,Φ )= ( Φ ( ˆΘ 2 β)φ T ˆΘ ) ˆΘ δ ˆΘ 2, if (iii) ˆΘ 2 β Φ T ˆΘ >. (21) Thi dpttion lw i identicl to (15) in ce (i) (ii), witche moothly to new expreion in ce (iii). The projection opertor Proj( ˆΘ,Φ ) i loclly Lipchitz in ( ˆΘ,Φ ), thu the ytem h unique olution defined for ome time intervl [,T), T >. Propoition 1: Under Aumption 1,2, 3, the control lw Eq. (9) long with the dpttion lw Eq. (14) (19) yield V T K d, (22) ˆΘ (t = ) Ω Θ ˆΘ (t) ˆΩ Θ, t. (23) Proof: The proof i trightforwrd therefore omitted here. It i imilr with the proof in the uthor previou work [11], [13]. From Propoition 1, one cn conclude tht, uing the feedbck control lw (9) the dpttion lw (14) (19), q t (D n + ˆD ) will not loe rnk, if 1 The dpttion lw i, in fct, only Lipchitz continuou, not continuouly differentible. The ue of the term mooth i light minomer in thi context, but we ue it here in ccordnce to prior uge in the literture. It hould be noted tht new prmeter projection opertor which i C n h been recently introduced in Ref. [12]. we chooe the initil prmeter gue ˆΘ () inide the et Ω Θ. For intnce, we my tke ˆΘ () =. It i lo worth mentioning tht the propoed dpttion lw (19) h the dditionl benefit of keeping the prmeter etimte from burting, which my hppen when the peritency of excittion condition doe not hold [14]. III. APPLICATION TO SPACECRAFT ATTITUDE CONTROL A. Eqution of Motion In thi ection, pplying the propoed dptive control cheme, we deign n dptive ttitude trcking control lw for pcecrft. A cluter of Vrible-Speed Control Moment Gyro (VSCMG) with N flywheel i ued for the torque ctutor. While conventionl Control Moment Gyro (CMG) keep it flywheel pinning t contnt rte, VSCMG it nme implie i eentilly ingle-gimbl CMG with the flywheel llowed to hve vrible peed. (See Ref. [1], [15] for more detil ppliction of VSCMG.) Inertil Frme Fig. 1. b 3 Body Frme g i i O b 1 b 2 Gimbl Frme Spcecrft Body with the i-th VSCMG. Figure 1 how pcecrft with the i-th VSCMG, g i i (body-fixed) gimbl xi, i i pin xi, t i g i i i trnvere torque xi. The eqution of motion of pcecrft with VSCMG re complicted hown in the forementioned reference, but under umption which re trd in the literture [11], [13], they cn be implified follow: J ω [h ]ω + Qu =, (24) h = Jω + A I w Ω, (25) Q = [Q cmg,q rw ] R 3 2N, (26) Q cmg (γ,ω) = A t I w Ω d, Q rw (γ) = A I w, the control input of thi ytem i u = [u 1,,u 2N ] T = [ γ T, Ω T ] T R 2N. (27) J i the totl moment of inerti of the pcecrft which i umed to be contnt 2, ω i the body rte vector of the pcecrft, h i the totl ngulr momentum of the pcecrft, γ = [γ 1,...,γ N ] T R N Ω = [Ω 1,...,Ω N ] T R N re vector of gimbl ngle flywheel pinning peed, 2 In fct, J i function of γ but the dependence i wek in generl. t i 2971

4 repectively, I w i digonl mtrix with the inerti of VSCMG flywheel. The kew-ymmetric mtrix [v ], for v R 3, repreent the cro product opertion. The mtrice A R 3 N hve column the gimbl (g i ), pin ( i ) trnvere (t i ) directionl unit vector expreed in the bodyfrme, i g, or t. Thee mtrice depend on the gimbl ngle follow A g = A g (28) A = A [coγ] d + A t [inγ] d (29) A t = A t [coγ] d A [inγ] d (3) the A denote the vlue of A t γ =. The ymbol x d denote the digonl mtrix with element the component of the vector x, coγ [coγ 1,,coγ N ] T inγ [inγ 1,,in γ N ] T. The modified Rodrigue prmeter (MRP) [16] [18] re choen to decribe the ttitude kinemtic of the pcecrft. 3 The kinemtic in term of the MRP i given by σ = G(σ)ω, (31) G(σ) = 1 (I 3 +[σ ]+σσ T [ 1 ) 2 2 (1+σT σ)]i 3 (32) I r i the r r identity mtrix. A uggeted in Ref. [1] [5], we combine the kinetic eqution (24) the kinemtic eqution (31) into one econd-order ytem follow: H(σ) σ +C(σ, σ) σ = D(σ)u (33) H(σ) = G T JG 1, (34) C(σ, σ) = G T JG 1 ĠG 1 G T [(R(σ)h I ) ]G 1, (35) D(σ) = G T Q. (36) R(σ) i rottionl mtrix from the inertil frme to the body frme, h I i the totl ngulr momentum of pcecrft expreed in the inertil frme which i conerved to be contnt if there i no externl torque pplied to the pcecrft. Therefore, h = R(σ)h I. Notice tht the eqution of motion (33) h the form of (1) with the grvittionl term g =. Moreover, it cn be eily hown tht the mtrix Ḣ 2C i kew-ymmetric [1]. B. Adptive Attitude Trcking Control Suppoe tht there re uncertintie in D well J. We ume tht the exct vlue of the initil xi direction of VSCMG ctutor t γ = re unknown. Thi cn hppen when the VSCMG re intlled with mll milignment /or the meure of gimbl ngle h contnt unknown bi. In ddition, h I i lo unknown contnt not only becue of uncertin J but lo becue of uncertin A. 3 We hten to point out tht the ue of the MRP to decribe the kinemtic i done without lo of generlity. Any other uitble kinemtic decription could hve been ued with the concluion of the pper remining eentilly the me. For mot ce the effect of xe uncertintie on the overll ytem performnce i not ignificnt. However, for the ce of flywheel ued mechnicl btterie in n Integrted Power Attitude Control Sytem (IPACS) [5], [19], [2], even mll milignment of the flywheel xe cn be detrimentl. Flywheel for IPACS ppliction pin t high peed hve lrge mount of tored kinetic energy ( hence ngulr momentum). Precie ttitude control require proper momentum mngement, while minimizing puriou output torque. Thi cn be chieved with the ue (in the implet cenrio) of t let four flywheel, whoe ngulr moment hve to be cnceled or regulted with high preciion. If the exct direction of the xe (hence the direction of the ngulr moment) re not known with ufficient ccurcy, lrge output torque error will impct the ttitude of the pcecrft. The uncertin prmeter in H C cn be defined follow: Θ = [ j 11, j 22, j 33, j 12, j 13, j 23, h 1, h 2, h 3 ] T R 9 (37) j re the element of J J J n J n i the nominl vlue of the ctul inerti mtrix J. Similrly, h re element of h I h I h In. The uncertin prmeter in D re defined Θ = [Θ T t,1,,θt t,n,θt,1,,θt,n ]T R 6N (38) Θ t,i t i, t n i,, Θ,i i, n i,, i = 1,,N. (39) t i, i, re ctul vlue of t i i t γ i =, repectively, t n i, n i, re their nominl vlue. The totl number of prmeter etimte i then 6N + 9. If one ue the method in the previou work [7], [1], [11], [13], then the number of the etimte will be much 6 (6N + 3). Exct mthemticl expreion of the regreor vector Y defined in (11) cn be eily obtined uing ymbolic mth pckge, cn be contructed from the meurement of σ, σ, the deired trjectorie σ d, σ d, σ d. The regreor vector Y defined in (12) cn be obtined in the me wy, but it i lo poible to derive it mthemticl expreion by mnipulting the mtrice follow: Y = T G T [(Mu) 1 I 3,,(Mu) 2N I 3 ] (4) [ [coγ] d [inγ] M d [inγ] d [coγ] d ][ Iw Ω d I w ] (41) (x) i i the ith element of vector x. Then uing the developed control cheme in Sec. II, one cn deign n dptive ttitude trcking control lw for the pcecrft. IV. NUMERICAL EXAMPLES Numericl exmple for tellite with VSCMG cluter re provided in thi ection to tet the propoed dptive control lgorithm. A trd four-vscmg pyrmid configurtion (N = 4) i utilized [2]. The prmeter ued for 2972

5 TABLE I SIMULATION PARAMETERS AND GAINS Symbol Vlue Unit σ() [,,] T ω() [,,] T rd/ec ω() [,,] T rd/ec 2 γ() [,,,] T rd γ() [,,,] T rd/ec Ω() 1 4 [2.5,3.5,3.5,3.] T rpm I w dig{2., 2., 2., 2.} kgm 2 K d 1 3 I 3 Λ I 3 β.1 δ.1 the imultion re hown in Tble I. Notice tht the initil wheel peed of the VSCMG re et to 25, 35, RPM, which re n order of mgnitude lrger thn the peed of conventionl CMG, ince the flywheel of VSCMG ued for IPACS in generl need to pin very ft o tht they re competitive gint trditionl chemicl btterie. According to Ref. [21], even higher peed thn thee vle i implementble, t let in lbortory. The nominl vlue of the xi direction t γ = [,,,] T re At n = A n = , (42). (43) The (unknown) ctul xi direction t γ = ued in the preent exmple re umed A = , (44) A t = , (45) which re obtined by rotting ech nominl gimbl frme (g i, i,t i ) with 1 degree bout rbitrry direction. With thee vlue, Θ The nominl vlue of the pcecrft inerti mtrix i J n = kgm 2 (46) the (unknown) ctul inerti mtrix i J = kgm 2 (47) which i obtined by dding/ubtrcting 2% of the nominl vlue. The reference trjectory i choen o tht the initil reference ttitude i ligned with the body frme which i lo ligned with the inertil frme, the ngulr velocity of the reference ttitude i choen ( ω d (t) =.4 in 2πt 4,in 2πt 3,in 2πt 2 ) T rd/ec. (48) Firt, in order to how the effect of the milignment of the xe of the VSCMG cluter the uncertin inerti mtrix, imultion without dpttion w performed. Figure 2 how the ttitude trcking error (expreed with Euler ngle) 4 under control lw with the dpttion gin Γ Γ et to zero mtrice. Since the flywheel peed re very ft, there i lrge ttitude trcking error without dpttion. Next, nother imultion w run with dpttion of the ctutor uncertinty Θ only. The dpttion gin re et to Γ = 1 I 6N Γ =, the reulting ttitude error i hown in Fig. 3. There i ignificnt performnce improvement with the dpttion of Θ only, but trcking error with mgnitude of bout.2 degree remin. On the other h, Fig. 4 how the ttitude trcking error with dpttion of Θ only. The dpttion gin re Γ = Γ = dig(1 7 I 6,1 5 I 3 ). In fct, the control lw in thi cenrio i lmot identicl with Slotine method [1]. The ttitude error i ignificntly ttenuted uing the dptive controller, but there re gin reidul trcking error with mgnitude of bout.1 degree. Finlly, imultion i performed with dpttion of both Γ Γ, tht i Γ = 1 I 6N Γ = dig(1 7 I 6,1 5 I 3 ). Figure 5 how the trcking performnce i improved upon Slotine control lw. Figure 6 how the time hitory of ˆΘ 2. It i confirmed tht ˆΘ 2 doe not drift more thn β + δ =.2 owing to the mooth projection lgorithm. A reult, the teering lw (9) remin well-defined. V. CONCLUSIONS In thi pper, we propoed n dptive trcking control lw for nonliner Hmiltonin MIMO dynmic ytem. The propoed control cheme h everl ignificnt improvement over the previou work in the literture. Firt, the propoed method cn del with uncertintie in the ctutor term which Slotine method [1] doe not del with. The regreor mtrix for dpttion of the uncertin inerti lo h mller ize thn tht of [1]. Second, the propoed method exploit the phyicl propertie of the Hmiltonin ytem o the deigned lw i more compct thn thoe in [7], [11], [13] which del with ctutor uncertintie but re derived bed on purely mthemticl pproche. Finlly, the propoed method del with the ctutor uncertintie eprtely from the uncertin inerti/corioli/grvity term. Therefore, it doe not need over-prmeteriztion to del with both kind of uncertintie t the me time, while [7], [11], [13] do. The developed dptive lgorithm i hown to ignificntly improve the trcking performnce in the ppliction to the 4 The ue of Euler ngle in the figure in done olely for the convenience of the reder who my not be fmilir with the MRP. 2973

6 pcecrft ttitude control, but it till h room for improvement. For intnce, while only 3 prmeter re generlly needed to expre milignment of xi frme for one VSCMG, totl of 6 prmeter re ued in thi method. Development of method to reduce the number of etimted prmeter would be extremely beneficil. REFERENCES [1] J. Slotine W. Li, Applied Nonliner Control. New Jerey: Prentice Hll, [2] F. L. Lewi, C. T. Abdllh, D. M. Dwon, Control of Robot Mnipultor. New York, NY: Mcmilln Publihing Compny, [3] J. J. E. Slotine M. D. D. Benedetto, Hmiltonin dptive control of pcecrft, IEEE Trnction on Automtic Control, vol. 35, no. 7, pp , July 199. [4] T. I. Foen, Comment on hmiltonin dptive control of pcecrft, IEEE Trnction on Automtic Control, vol. 38, no. 4, pp , April [5] H. Yoon P. Tiotr, Spcecrft dptive ttitude power trcking with vrible peed control moment gyrocope, Journl of Guidnce, Control, Dynmic, vol. 25, no. 6, pp , Nov.-Dec. 22. [6] S. S. Ge, Adptive control of robot hving both dynmicl prmeter uncertintie unknown input cling, Mechtronic, vol. 6, no. 5, pp , [7] Y.-C. Chng, An dptive H trcking control for cl of nonliner multiple-input-multiple-output (MIMO) ytem, IEEE Trnction on Automtic Control, vol. 46, no. 9, pp , Sep. 21. [8] J.-B. Pomet L. Prly, Adptive nonliner regultion : Etimtion from the Lypunov eqution, IEEE Trnction on Automtic Control, vol. 37, no. 6, pp , June [9] H. K. Khlil, Adptive output feedbck control of nonliner ytem repreented by iuput-output model, IEEE Trnction on Automtic Control, vol. 41, no. 2, pp , Feb [1] H. Yoon, Spcecrft ttitude power control uing vrible peed control moment gyro, Ph.D. dierttion, Georgi Intitute of Technology, Atlnt, GA., December 24, (URL: [11] H. Yoon P. Tiotr, Adptive pcecrft ttitude trcking control with ctutor uncertintie, Journl of the Atronuticl Science, ubmitted for publiction. [12] Z. Ci, M. de Queiroz, D. Dwon, A ufficiently mooth projection opertor, IEEE Trnction on Automtic Control, vol. 51, no. 1, pp , Jn. 26. [13] H. Yoon P. Tiotr, Adptive pcecrft ttitude trcking control with ctutor uncertintie, in AIAA Guidnce, Nvigtion, Control Conference, Sn Frncico, Cliforni, Augut 25, IAA [14] B. D. O. Anderon, Filure of dptive control theory their reolution, Communiction in Informtion Sytem, vol. 5, no. 1, pp. 1 2, 25. [15] H. Schub, S. R. Vdli, J. L. Junkin, Feedbck control lw for vrible peed control moment gyrocope, Journl of the Atronuticl Science, vol. 46, no. 3, pp , [16] P. Tiotr, Stbiliztion optimlity reult for the ttitude control problem, Journl of Guidnce, Control, Dynmic, vol. 19, no. 4, pp , [17] H. Schub J. L. Junkin, Stereogrphic orienttion prmeter for ttitude dynmic: A generliztion of the Rodrigue prmeter, Journl of the Atronuticl Science, vol. 44, no. 1, pp. 1 19, [18] M. D. Shuter, A urvey of ttitude repreenttion, Journl of the Atronuticl Science, vol. 41, no. 4, pp , [19] P. Tiotr, H. Shen, C. Hll, Stellite ttitude control power trcking with energy/momentum wheel, Journl of Guidnce, Control, Dynmic, vol. 24, no. 1, pp , 21. [2] H. Yoon P. Tiotr, Singulrity nlyi of vrible-peed control moment gyro, Jornl of Guidnce, Control, Dynmic, vol. 27, no. 3, pp , 24. [21] R. Hebner, J. Beno, A. Wll, Flywheel btterie come round gin, IEEE SPECTRUM, vol. 39, pp , Apr Fig. 2. Fig. 3. Fig Trcking Error Without Adpttion Trcking Error With Adpttion of Θ Only Trcking Error With Adpttion of Θ Only Fig. 5. Trcking Error With Adpttion of Θ Θ Squre of Norm of Prmeter Etimte Fig. 6. Squre of Norm of Prmeter Etimte ˆΘ. 2974